text: An inexact differential  is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths  such that ,  and
  In this case, we denote the integrals as  and  respectively to make explicit the path dependence of the change of the quantity we are considering as .
  More generally, an inexact differential  is a differential form which is not an exact differential, i.e., for all functions ,
  The fundamental theorem of calculus for line integrals requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative. This is because there can be no unique representation of an antiderivative for inexact differentials since their variation is inconsistent along different paths. This stipulation of path independence is a necessary addendum to the fundamental theorem of calculus because in one-dimensional calculus there is only one path in between two points defined by a function.